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Under suitable requirements on a kernel on a locally compact space, wedevelop a theory of inner (outer) balayage of quite general Radon measures$\omega$ (not necessarily of finite energy) onto quite general sets (notnecessarily closed). We prove the existence and the uniqueness of inner (outer)swept measures, analyze their properties, and provide a number of alternativecharacterizations. In spite of being in agreement with Cartan's theory ofNewtonian balayage, the results obtained require essentially new methods andapproaches, since in the case in question, useful specific features ofNewtonian potentials may fail to hold. The theory thereby established extendsconsiderably that by Fuglede (Anal. Math., 2016) and that by the author (Anal.Math., 2022), these two dealing with $\omega$ of finite energy. Such ageneralization enables us to improve substantially our recent results on theGauss variational problem (Constr. Approx., 2024), by strengthening theirformulations and/or by extending the area of their validity. This study coversmany interesting kernels in classical and modern potential theory, which alsolooks promising for other applications.

General theory of balayage on locally compact spaces. Applications to weighted minimum energy problems